{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0082",
  "pilot_number": "R0004-02632",
  "number": 2632,
  "family_id": "qf-app21-representation-consistency-audit",
  "title": {
    "zh-Hant": "第 2632 題：數學表徵語義橋",
    "en": "Question 2632: Audit Every Table Row against an Affine Rule"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r01-mathematical-representation-0082",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mathematical-representation"
    ],
    "title_zh": "數學表徵語義橋",
    "stem_zh": "明示文字規則、符號或有限表格。操作=representation-consistency-audit；資料={\"a\":-1,\"b\":-4,\"pairs\":[[0,-4],[1,-5],[2,-6]]}。請依 [a,b,rows,all_match] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-1, -4, [[0, -4, -4, 是], [1, -5, -5, 是], [2, -6, -6, 是]], 是)",
      "explanation_zh": "應用擴展核對=(-1, -4, [[0, -4, -4, 是], [1, -5, -5, 是], [2, -6, -6, 是]], 是)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；不由圖形外觀補值；有限表徵一致不證未列域上的等價。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "數學表徵語義橋"
      ]
    },
    "variant": {
      "family_id": "qf-app21-representation-consistency-audit",
      "seed": 202610211082,
      "parameters": {
        "a": -1,
        "b": -4,
        "pairs": [
          [
            0,
            -4
          ],
          [
            1,
            -5
          ],
          [
            2,
            -6
          ]
        ],
        "domain": "Finite_representation_bridge",
        "operation": "representation-consistency-audit",
        "semantic_template": "qf-app21-representation-consistency-audit/{\"a\":-1,\"b\":-4,\"pairs\":[[0,-4],[1,-5],[2,-6]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r01-mathematical-representation-0082",
    "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f765245d23de519f6cac01568d8a8dcf4190cec3d8220e4d3cba3a4b6fe248fc",
      "family_id": "qf-app21-representation-consistency-audit",
      "seed": 202610211082,
      "semantic_template": "qf-app21-representation-consistency-audit/{\"a\":-1,\"b\":-4,\"pairs\":[[0,-4],[1,-5],[2,-6]]}"
    },
    "fields": {
      "title": "Audit Every Table Row against an Affine Rule",
      "stem": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-4, pairs=[[0, -4], [1, -5], [2, -6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?).",
      "answer": {
        "kind": "expression",
        "value": "(-1, -4, [[0, -4, -4, Yes], [1, -5, -5, Yes], [2, -6, -6, Yes]], Yes)",
        "explanation": "Compare actual numeric entries with the rule rather than judging a plotted line’s appearance. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, -4, [[0, -4, -4, Yes], [1, -5, -5, Yes], [2, -6, -6, Yes]], Yes)."
      },
      "skills": [
        "Audit Every Table Row against an Affine Rule"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "明示文字規則、符號或有限表格。操作=representation-consistency-audit；資料={\"a\":-1,\"b\":-4,\"pairs\":[[0,-4],[1,-5],[2,-6]]}。請依 [a,b,rows,all_match] 欄序回答。",
        "en": "Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-4, pairs=[[0, -4], [1, -5], [2, -6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared symbolic rule or complete finite table",
      "after": "keep the stated domain and representation distinct",
      "relation": "implies",
      "rule_id": "representation_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只由明示規則或表格計算；有限表格的一致性不自動證明未列出論域上的函數等同。",
        "en": "Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "audit x=0 ax := mul(-1, 0)",
      "after": "audit x=0 ax = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=0 ax，不以小數近似替代。",
          "en": "Recompute audit x=0 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "audit x=0 ax"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "audit x=0 ax+b := sum(0, -4)",
      "after": "audit x=0 ax+b = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=0 ax+b，不以小數近似替代。",
          "en": "Recompute audit x=0 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-4"
        ],
        "output": "-4",
        "label": "audit x=0 ax+b"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "listed y=-4; rule ax+b=-4",
      "after": "agreement=是",
      "relation": "implies",
      "rule_id": "representation_row_agreement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐列比較已列輸出與規則代值，不以圖形看來接近代替精確相等。",
        "en": "Compare each listed output with the exact rule value;visual proximity is not equality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "audit x=1 ax := mul(-1, 1)",
      "after": "audit x=1 ax = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=1 ax，不以小數近似替代。",
          "en": "Recompute audit x=1 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "audit x=1 ax"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "audit x=1 ax+b := sum(-1, -4)",
      "after": "audit x=1 ax+b = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=1 ax+b，不以小數近似替代。",
          "en": "Recompute audit x=1 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-1",
          "-4"
        ],
        "output": "-5",
        "label": "audit x=1 ax+b"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "listed y=-5; rule ax+b=-5",
      "after": "agreement=是",
      "relation": "implies",
      "rule_id": "representation_row_agreement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐列比較已列輸出與規則代值，不以圖形看來接近代替精確相等。",
        "en": "Compare each listed output with the exact rule value;visual proximity is not equality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "audit x=2 ax := mul(-1, 2)",
      "after": "audit x=2 ax = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算乘法a·x，避免把a(x+b)誤作原規則。",
        "en": "Evaluate a·x first;do not replace the rule by a(x+b)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=2 ax，不以小數近似替代。",
          "en": "Recompute audit x=2 ax with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "audit x=2 ax"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "audit x=2 ax+b := sum(-2, -4)",
      "after": "audit x=2 ax+b = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加截距b，完整代入此輸入。",
        "en": "Then add the intercept b,fully substituting this input."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 audit x=2 ax+b，不以小數近似替代。",
          "en": "Recompute audit x=2 ax+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-4"
        ],
        "output": "-6",
        "label": "audit x=2 ax+b"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "listed y=-6; rule ax+b=-6",
      "after": "agreement=是",
      "relation": "implies",
      "rule_id": "representation_row_agreement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐列比較已列輸出與規則代值，不以圖形看來接近代替精確相等。",
        "en": "Compare each listed output with the exact rule value;visual proximity is not equality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "[是, 是, 是]",
      "after": "是",
      "relation": "implies",
      "rule_id": "all_representation_rows",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部列都符合才通過；任一反例就足以否定這張表與此規則的一致性。",
        "en": "All listed rows must agree;one counterexample refutes consistency of this table with the rule."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      -1,
      -4,
      [
        [
          0,
          -4,
          "-4",
          true
        ],
        [
          1,
          -5,
          "-5",
          true
        ],
        [
          2,
          -6,
          "-6",
          true
        ]
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(-1, -4, [[0, -4, -4, 是], [1, -5, -5, 是], [2, -6, -6, 是]], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261013-advance-20261021-r01-mathematical-representation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/questions.jsonl",
      "batch_sha256": "c3d5564b7f52318668739030fffceffb4febaa2d3d1229aa1ac4a96d58a778b8",
      "original_line": 82,
      "original_line_sha256": "75bf4a74d99e6648cd6c31fbcd829f9259a1ccf2daec2617f1777c16b04ebc6a",
      "question_compact_sha256": "f765245d23de519f6cac01568d8a8dcf4190cec3d8220e4d3cba3a4b6fe248fc",
      "english_row_compact_sha256": "607398507384a0c7e327ac3f1de5387b71613f344ea766df2d85a62c1cf353ca",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r01-mathematical-representation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-mathematical-representation": {
        "sha256": "4f70691fec7da4288ceee8ebb2ce291dfee4c28ac4a2e8aa851fa23fe5757fd5",
        "english_sha256": "6774f494e9469b407eedbc3e1c065e7a85ce40b95fbd32d15c7cb728f106fa88",
        "original_path": "public/data/mko/mathematical-representation.json",
        "english_path": "public/data/i18n/editions/mko-mathematical-representation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "92bef24b029015603ef957694e449abf34325c4e0eea23ccc4af5cc510d1254d",
    "original_candidate_line_sha256": "92bef24b029015603ef957694e449abf34325c4e0eea23ccc4af5cc510d1254d"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p5",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "355db5703280d6bfba055fb90de1cd4a560582332036ed9dbdeffcef20b54f6b",
      "arithmetic_steps": 6
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}